kernel(RingMap,ZZ) -- homogeneous components of the kernel of a homogeneous ring map

Synopsis

Function: kernel
Usage:

kernel(phi,d)
Inputs:
- phi, a ring map, $K[y_0,\ldots,y_m]/J \to K[x_0,\ldots,x_n]/I$, defined by homogeneous forms of the same degree and where $J$ and $I$ are homogeneous ideals
- d, an integer
Optional inputs:
- SubringLimit => ..., default value infinity,
Outputs:
- the ideal generated by all homogeneous elements of degree d belonging to the kernel of phi

Description

This is equivalent to ideal image basis(d,kernel phi), but we use a more direct algorithm. We take advantage of the homogeneity and reduce the problem to linear algebra. For small values of d this method can be very fast, as the following example shows.

i1 : phi = toMap map specialQuadraticTransformation 8

                                                                                              2                                     2                                                                                                                                                            2                             2                                                                                                                               2                              2                                                                                                                                                                                     2                                                                                                     2                            2                                                                                              2                           2                                           2                                                                   2           2
o1 = map (QQ[x ..x ], QQ[y ..y  ], {- 5x x  + x x  + x x  + 35x x  - 7x x  + x x  - x x  - 49x  - 5x x  + 2x x  - x x  + 27x x  - 4x  + x x  - 7x x  + 2x x  - 2x x  + 14x x  - 4x x , - x x  - 6x x  - 5x x  + 2x x  + x x  + x x  - 5x x  - x x  + 2x x  + 7x x  - 2x x  + 2x x  - 3x x , - 25x  + 9x x  + 10x x  - 2x x  - x  + 29x x  - x x  - 7x x  - 13x x  + 3x x  + x x  - x x  + 2x x  - x x  + 7x x  - 2x x  - 8x x  + 2x x  - 3x x , x x  + x x  + x  + 7x x  - 9x x  + 12x x  - 4x  + 2x x  + 2x x  - 14x x  + 4x x  + x x  - x x  - 14x x  + x x , - 5x x  + x x  - 7x x  + 8x x  - 5x x  + 2x x  - x x  + x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , x x  + x  - 7x x  - 8x x  + x x  + x x  + 2x x  - x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  + x  - 8x x  + x x  + 6x x  - 2x  + x x  + x x  - 7x x  + 2x x  + x x  - 7x x  + 2x x , x x  - 7x x  + x x  + x x  - 7x x  + 2x  - x x , - 4x x  + x x  - x  - 7x x  + 8x x  + x x  - x x  - 6x x  + 2x  - x x  - x x  + 7x x  - 2x x  - x x  + 7x x  - 2x x , - 5x x  + 2x  + x x  - x  - x x  + 8x x  - 10x x  + 2x x  + 2x x  - 2x x  + 14x x  - 4x x  + 5x x  - 3x x  - 2x x  + 7x x  - 2x x  - 3x x , - 5x x  + x x  + x x  - 4x x  - x x  + x x  + x x , x x  - x x  + 5x x  + x x  - 14x x  - x x  - 8x x  - 8x x  + 2x x  + 4x x  + 2x x  + 4x x  + 3x x  - 7x x  + 2x x  - 3x x })
              0   8       0   11        0 3    2 4    3 4      0 5     2 5    3 5    4 5      5     0 6     2 6    4 6      5 6     6    4 7     5 7     6 7     4 8      5 8     6 8     1 2     1 5     0 6     1 6    4 6    5 6     0 7    1 7     2 7     5 7     6 7     1 8     7 8       0     0 2      0 4     2 4    4      0 5    2 5     4 5      0 6     4 6    5 6    0 7     2 7    4 7     5 7     6 7     0 8     4 8     7 8   2 4    3 4    4     2 5     4 5      5 6     6     3 7     4 7      5 7     6 7    3 8    4 8      5 8    6 8      0 4    2 4     2 5     4 5     0 6     2 6    4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   0 4    4     1 5     4 5    0 6    1 6     4 6    5 6    4 7     5 7     6 7    4 8     5 8     6 8   2 3    4     4 5    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8   1 3     1 5    1 6    4 6     5 6     6    3 7      0 3    3 4    4     0 5     4 5    0 6    4 6     5 6     6    3 7    4 7     5 7     6 7    4 8     5 8     6 8      0 2     2    2 4    4    2 5     4 5      0 6     5 6     2 7     4 7      5 7     6 7     0 8     2 8     4 8     5 8     6 8     7 8      0 1    1 2    1 4     0 6    1 6    4 6    0 7   0 2    1 2     0 4    1 4      1 5    2 5     4 5     0 6     1 6     4 6     2 7     0 8     1 8     5 8     6 8     7 8

o1 : RingMap QQ[x ..x ] <-- QQ[y ..y  ]
                 0   8          0   11

i2 : time kernel(phi,1)
 -- used 0.0770821s (cpu); 0.0360112s (thread); 0s (gc)

o2 = ideal ()

o2 : Ideal of QQ[y ..y  ]
                  0   11

i3 : time kernel(phi,2)
 -- used 0.339101s (cpu); 0.255933s (thread); 0s (gc)

                           2                                                
o3 = ideal (y y  + y y  + y  + 5y y  + y y  + 5y y  - y y  - 4y y  - 5y y  -
             2 4    3 4    4     2 5    3 5     4 5    1 6     2 6     5 6  
     ------------------------------------------------------------------------
                                                                           
     4y y  - 2y y  - y y  + 4y y  - 5y y  - 4y y  + 3y y  - 4y y  - y y   -
       2 7     4 7    1 8     4 8     5 8     5 9     7 9     8 9    3 10  
     ------------------------------------------------------------------------
                                                                        2
     3y y   - 5y y   - y y   + 4y y   + 5y y  , 3y y  - y y  - 3y y  - y  +
       6 10     8 10    4 11     6 11     8 11    1 3    2 3     3 4    4  
     ------------------------------------------------------------------------
     2y y  - y y  + y y  + 2y y  + 3y y  - 7y y  - 4y y  + 7y y  - 2y y  +
       0 5    3 5    1 6     2 6     5 6     2 7     4 7     1 8     4 8  
     ------------------------------------------------------------------------
     y y  - y y  + 2y y  + 2y y  + y y  - 7y y   + 5y y   - 3y y   - y y   -
      0 9    4 9     5 9     7 9    8 9     0 10     3 10     6 10    0 11  
     ------------------------------------------------------------------------
     2y y   - 2y y  , 7y y  + y y  + 7y y  - y y  + 8y y  - y y  - y y  +
       3 11     4 11    0 1    0 4     1 4    3 4     0 5    3 5    1 6  
     ------------------------------------------------------------------------
     7y y  + 8y y  + y y  + 8y y  - y y  - 8y y  + 7y y  - 8y y  + 7y y  +
       2 6     5 6    2 7     4 7    1 8     4 8     5 9     7 9     8 9  
     ------------------------------------------------------------------------
     y y   - y y   + 8y y   - 7y y   - 7y y   - 7y y  )
      0 10    3 10     6 10     0 11     4 11     6 11

o3 : Ideal of QQ[y ..y  ]
                  0   11

Ways to use this method: